Advanced topics in algebra: Algebraic geometry

1. How to write a clear math paper by Igor Pak.
2. Mathematical Writing by Donald Knuth et al.

The format of our classwork will be as follows. The first meeting of the week (Monday) will be a traditional lecture in which I will explain the key ideas of a topic, but not prove all the details. In the following two meetings (Wednesday, Thursday), we will work out the details in small groups. By the end of Thursday, you will write a “progress report” (a 1-2 sentence summary of your progress for each question) and submit it on Gradescope. On Friday, based on the progress reports, I will present the solutions. Students (selected at random) will write up the solutions and send them to me in a week. I will add the write-ups to our notes.

I will not accept any late submissions, except for medical emergencies with a medical certificate. To compensate for this strict policy, I will drop the lowest among the (homework+classwork) marks.

You are allowed, even encouraged, to work with others on homework assignments, but you must write up your solutions on your own. In other words, you may not copy someone else’s write-up and you may not write your solutions side by side with someone else. On your submission, you must write the names of your collaborators. This is a matter of academic honesty; it will not affect your marks.

There will be no collaboration on the exams.

1. A homogeneous polynomial in variables \(X_1, \dots, X_n\) is one whose every (non-zero) term has the same total degree. For example, \(X_1^2 + X_2X_3\) is homogeneous (of degree 2), whereas \(X_1 + X_2X_3\) is not homogeneous. Every polynomial \(p\) can be written uniquely as \(p = p_0 + p_1 + \dots + p_d\), where \(p_i\) is homogeneous of degree \(i\). The \(p_i\)’s are called the homogeneous components of \(p\).

   Let \(k\) be a field and let \(I \subset k[X_1, \dots, X_n]\) be an ideal. Prove that the following are equivalent:

   1. \(I\) is generated by homogeneous polynomials.
   2. For every \(p\) in \(I\), all the homogeneous components of \(p\) are also in \(I\).

   Definition: An ideal satisfying the above conditions is called a homogeneous ideal.

2. Let \(k\) be an infinite field. Let \(X \subset \mathbb A^n_k\) be a Zariski closed subset. Prove that the following are equivalent:
   1. \(I(X)\) is a homogeneous ideal.
   2. For every \(x\) in \(X\) and \(\lambda\) in \(k\), the scalar multiple \(\lambda \cdot x\) is also in \(X\).

3. Let \(k\) be an algebraically closed field and let \(X \subset \mathbb A_k^2\) be the subset defined by the (infinite) system of equations

   \begin{align*} x + y &= 0 \\ x^2 + y^2 &= 0\\ x^3 + y^3 &= 0\\ \cdots \end{align*}

   Find out a finite system of polynomial equations that defines \(X\), or prove that such a system does not exist.

   Caution: The answer may depend on the characteristic of \(k\).

4. Let \(P_n\) denote the set of monic polynomials in one variable \(T\) with coefficients in \(\mathbb C\). Identify the set \(P_n\) with the affine space \(\mathbb A_{\mathbb C}^n\) by the rule \[T^n + a_{n-1}T^{n-1} + \cdots + a_1 T + a_0 \leftrightarrow (a_0, \dots, a_{n-1}).\] Prove that the set of polynomials with \(n\) distinct roots is a Zariski open subset of \(\mathbb A^n_{\mathbb C}\). Describe the equations that define its complement when \(n = 2\).

   Hint: Remember `the discriminant’ from Algebra 2.

5. Describe all maximal ideals of the following rings
   1. \(\mathbb C[x,y,z]/(xy, yz, xz)\)
   2. \(\mathbb R[x,y]\)

1. Let \(\phi \colon X \to Y\) be a regular map between quasi-affine varieties. Prove that \(\phi\) is continuous in the Zariski topology.
2. Let \(U \subset V \subset \mathbb A^n\) be two non-empty open sets. Show that the map \(k[V] \to k[U]\), obtained by restriction, is injective. As a result, show that we have an injective map \[ k[V] \to k(x_1,\dots,x_n).\]
3. Let \(f \in k[x_1,\dots,x_n]\) be non-zero, and let \(U_f\) be the complement in \(\mathbb A^n\) of \(V(f)\). Identify the ring \(k[U]\) as a sub-ring of the fraction field \(k(x_1,\dots,x_n)\).
4. Let \(U = \mathbb A^2 \setminus \{(0,0)\}\). Show that the restriction map \(k[x,y] \to k[U]\) is an isomorphism. Hint: Observe that \(U\) is the union \(U_x \cup U_y\).
5. Let \(S \subset \mathbb P^3\) be the Fermat cubic \[ S = V(X^3+Y^3+Z^3+W^3). \] Let \(L \subset S\) be the line defined by \[ L = V(X+Y, Z+W).\] Consider the regular map \[ \phi \colon S \setminus L \to \mathbb P^1\] defined by \[ [X:Y:Z:W] \mapsto [X+Y:Z+W].\] Prove that \(\phi\) extends to a regular map \[ \phi \colon S \to \mathbb P^1.\]

Not to be turned in, but highly recommended.

What are the fibers of the map \(\phi \colon S \to \mathbb P^1\)? Can you picture this over the real numbers?

1. Given 3 distinct points \(p, q, r \in \mathbb P^1\), prove that there exists a unique element of \(A \in \operatorname{PGL}_2(k)\) such that \[ A [0:1] = p, \quad A[1:0] = q, \quad A[1:1] = r.\] What is the analogous statement for \(\mathbb P^n\)? Just write the statement, not the proof.
2. Let \(Y\) be a separated variety, and \(U \subset X\) a dense subset. If two regular maps \(f, g \colon X \to Y\) agree on \(U\), then show that they must agree on \(X\). In other words, if \(Y\) is separated, then a continuous map \(U \to Y\) has at most one extension \(X \to Y\).
3. Prove that any rational map \(\mathbb P^1 \dashrightarrow \mathbb P^n\) extends to a regular map.
4. Consider \(\mathbb A^2\) as an open subset of \(\mathbb P^2\) in the standard way: \[ \mathbb A^2 = \{[x:y:1] \mid x, y \in k\}.\] For \(f \in k[x,y]\), consider \(C = V(f) \subset \mathbb A^2\). The closure of \(C\) in \(\mathbb P^2\) has the form \(V(F)\) for some homogeneous polynomial \(F \in k[X,Y,Z]\). Describe how to obtain \(F\) from \(f\) and prove that \(V(F)\) is indeed the closure of \(V(f)\).

1. Consider the \(\mathbb A^n\) of monic polynomials of degree \(n\) in one variable. Given \(r \in \{1,\dots,n\}\), let \(X_r \subset \mathbb A^n\) be the set of polynomials that have a root of order at least \(r\). It turns out that \(X_r \subset \mathbb A^n\) is a Zariski closed subset. Prove that \(X_r\) is irreducible.

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2. Let \(X\) and \(Y\) be two irreducible varieties and \(\phi \colon X \dashrightarrow Y\) a birational isomorphism. Prove that there exist non-empty open subsets \(U \subset X\) and \(V \subset Y\) such that the map \(\phi\) induces an isomorphism \(\phi \colon U \to V\). Identify such open subsets \(U\) and \(V\) for the Cremona transformation \[ \phi \colon \mathbb P^2 \dashrightarrow \mathbb P^2 \] defined by \[ \phi \colon [X:Y:Z] \mapsto [1/X:1/Y:1/Z].\]

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3. It turns out that every smooth cubic surface \(S \subset \mathbb P^3\) is birational to \(\mathbb P^2\). One proof goes as follows. First, we prove that \(S\) contains two skew (non-intersecting) lines \(L\) and \(M\). We then define a rational map \[\phi \colon L \times M \dashrightarrow S\] by mapping \((x,y)\) to the unique third point of intersection of the line joining \(x\) and \(y\) with the cubic \(S\).

   Think (but do not turn in): why should \(\phi\) be a birational isomorphism? What should be the inverse map?

   Consider the specific cubic surface \[ S = V(X^3+Y^3+Z^3+W^3) \subset \mathbb P^3.\] Find two skew lines on \(S\) and using them write down the map \(\phi\).

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4. Fix a positive integer \(d\) and let \(V\) be the vector space of homogeneous polynomials of degree \(d\) in \(X, Y, Z\). Each point of \(\mathbb P V\) represented, say, by an \(F \in V\) defines a plane curve \(V(F) \subset \mathbb P^2\). So we can view \(\mathbb P V\) as a parameter space for all plane curves.

   Given \([F] \in \mathbb P V\) and \(p \in \mathbb P^2\), we say that \(F\) has a singularity at the point \(p\) if all three partials \(\frac{\partial F}{\partial X}\), \(\frac{\partial F}{\partial Y}\), \(\frac{\partial F}{\partial Z}\) vanish at \(p\). Let \(\Delta \subset \mathbb P V\) be the set of \([F]\) which have a singularity at some point. It turns out that \(\Delta\) is Zariski closed and irreducible (we will be able to prove this soon). Find the dimension of \(\Delta\).

   Hint: Do a dimension count using \(\{(F,p) \mid F \text{ is singular at } p\} \subset \mathbb P V \times \mathbb P^2\). You may use without proof that this space is irreducible.

Look up the definition of an algebraic group or a group variety; examples include \({\rm GL}_n\), \({\rm SL}_n\), \({\rm O}_n\), \({\rm SO}_n\), and so on. An algebraic action of an algebraic group \(G\) on an algebraic variety \(X\) is a regular map \(\alpha \colon G \times X \to X\) satisfying the usual properties of a group action. We usually denote \(\alpha(g,x)\) by \(g \cdot x\).

In the following, all varieties are assumed to be separated.

1. Suppose \(\alpha \colon G \times X \to X\) is an action of algebraic group \(G\) on an algebraic variety \(X\). Given \(x \in X\), the stabiliser of \(x\) is the subset \(G_x \subset G\) consisting of \(g \in G\) such that \(g \cdot x = x\). Prove that \(G_x \subset G\) is Zariski closed.
2. Let \(O_x \subset X\) be the orbit of \(x\). Assume that \(G\) is irreducible and prove that \(O_x \subset X\) is irreducible.
3. Show by an example that the orbit may not be closed. (But they always turn out to be locally closed; you do not have to prove this.)
4. Assuming \(G\) and \(G_x\) are irreducible, prove the orbit-stabiliser theorem \[ {\rm dim}\ O_x + {\rm dim}\ G_x = {\rm dim}\ G.\]

5. Compute the tangent space at the identity matrix of the following algebraic groups (all subgroups of \({\rm GL_n}\)):

   1. \({\rm GL}_n\).
   2. \({\rm SL}_n = \{A \mid \det A = 1\}\).
   3. \({\rm O}_n = \{A \mid A A^T = {\rm id}\}\).

   Your answer must be of the form: the tangent space consists of \(n \times n\) matrices \(M\) of the form ….